The following are four different relations about displacement, velocity and acceleration for the motion of a particle in general.
 

(a) \(v_{a v}=1 / 2\left[v\left(t_1\right)+v\left(t_2\right)\right]\)
(b) \(v_{\mathrm{av}}=\mathrm{r}\left(\mathrm{t}_2\right)-\mathrm{r}\left(\mathrm{t}_1\right) / \mathrm{t}_2-\mathrm{t}_1\)
(c) \(r=1 / 2\left[v\left(t_2\right)-v\left(t_1\right)\right]\left(\mathrm{t}_2-\mathrm{t}_1\right)\)
(d) \(\mathrm{a}_{\mathrm{av}}=v\left(\mathrm{t}_2\right)-v\left(\mathrm{t}_1\right) / \mathrm{t}_2-\mathrm{t}_1\)

The incorrect alternative/s is/are:

1. (a, d)
2. (a, c)
3. (b, c)
4. (a, b)

(2) Hint: Recall the concept of average velocity and average acceleration.

Step 1: Find the average velocity of the object.

If an object undergoes a displacement Ar in time At, its average velocity is given by
v=ΔrΔt=r2r1t2t1 where r1 and r2 are position vectors corresponding to time t, and t

Step 2: Find the average acceleration.
t the velocity of an object changes from v, to v in time At. Average acceleration is given by
aav=ΔvΔt=v2v1t2t2
But, when acceleration is non-uniform,

                         vav  v1 + v22We can write    Δv = ΔrΔtHence,               Δr = r2  r1=(v2  v1)(t2  t1)